Pages

Sunday, 8 September 2013

Quantum Circuits

Like classical circuits, quantum circuits can be thought of as directed acyclic graphs. A register of qubits refers to a collection of qubits like the classical bitstring. A quantum circuit
can consist of one or more registers which partition the qubits. The
nodes of the graph represent unitary quantum gates. Starting from an
initial state which represents the input, the qubits are fed through
quantum gates. If gates are to be applied sequentially on some qubits,
then the overall action of this sequence of gates is given by the
ordinary matrix product of the individual gates in the same sequential
order. If quantum gates are to be applied to different parts of the
register in parallel, then the tensor product of these gates gives the
overall action on the registers.

A significant difference for
quantum circuits is the necessity of performing a measurement on all or
some of the qubits at the end of a computation to yield an output
corresponding to some computational basis state. Since the measurement
process is inherently probabilistic, it follows that the model of
quantum computation is itself a probabilistic model. Obviously, quantum
circuits do differ from classical circuits, but it can be shown that
quantum circuits can simulate the action of any classical circuit and
vice versa, but going in the latter direction may require substantially
more resources or gates.

In summary, the objective of quantum
computation is to start with some finite number of qubits. Then apply
quantum gates to manipulate the qubits in such a way so that upon
measurement a meaningful output state yielding a correct solution is
observed with high probability.